Steven H. Collicott

Equations of Motion

This chapter is the first of three which set out the fundamental fluid dynamics required for the further development of aerodynamics. In this chapter the study of air in motion starts with the physics and mathematics of one-dimensional fluid motion. Many of the physical phenomena evident in all stages of aerodynamics are most readily approached by considering the one-dimensional manifestation of the phenomena. The physical laws governing the changes in the physical properties of air are first covered and the relevant mathematics introduced. These physical laws are applied to the accelerating gas as it moves out of the low-speed (incompressible) regime and into the transonic and supersonic regimes where the abrupt changes in properties are manifest.

Chapter 4 – Compressible Flow

This chapter is the third of three which set out the fundamental fluid dynamics required for the further development of aerodynamics. Compressible flows physics are introduced with quasi-one-dimensional (or Q1D flow), an approximation good for many nozzles and for external aerodynamics. Quasi-one dimensional flows are used as the framework to develop and discuss the governing equations for one-dimensional compressible flow. Steady normal shock waves are introduced in supersonic one-dimensional compressible flow. The effect of normal shocks on thermodynamic, flow, and stagnation properties of the gas are studied in detail as they are highly relevant to more complex shock waves. The formation of Mach and shock waves in two-dimensional flows and how the flow is affected by such waves is described.

Chapter 3 – Viscous Flow and Boundary Layers

This chapter is the second of three which set out the fundamental fluid dynamics required for the further development of aerodynamics. This chapter deals with the analysis of viscous flows and viscous drag. Powerful drivers for most every wing design effort are maximum lift and minimum drag. Learning to meet both of these goals in a dependable, practical, and affordable manner is a very advanced topic, so in this chapter a brief history of previous attempts to maximizing lift and minimizing drag is presented. Many items in this history remain in use today in aircraft and can drive the creative thinking of designers for new aircraft. Critical to almost every practical lift and drag improvement are the boundary layers on the vehicle, especially the boundary layers on the wing. So boundary layers are reviewed from a practical point of view. Boundary layer transition, separation, and manipulation (flow control) are presented as an overview rather than as a complete derivation.

Chapter 10 – Flow Control and Wing Design

This chapter is an overview of the problems confronted by the aeronautical engineer when designing a wing system including the devices used to control flows as required by certain operating conditions.

Chapter 5 – Potential Flow

This chapter is an introduction to potential-flow theory as applied to calculate the air flow and pressure distribution around various shapes of body. The classical assumption of incompressible, irrotational and inviscid flow and its meaning that the vorticity is everywhere zero is examined in some detail. The flows described by this model are potential fields. The velocity potential function and the determination of the velocity components from this scalar function are described. A description of the reduction of the equations of motion for ‘ideal’ (irrotational, incompressible and inviscid) flow to a single equation, viz., the Laplace equation, is provided. The classical analytical techniques are applied to examine two-dimensional and axisymmetric solutions to the Laplace equation for aerodynamic applications. The uniform stream solution and the singular solutions known as source, doublet and vortex are examined and applied to construct simple bodies. These include the flow around the Rankine leading edge, the Rankine oval, the circular cylinder with and without circulation. The pressure distributions and the force acting on these bodies are examined. Computational tools that may be applied to predict potential flows around arbitrary two-dimensional geometries are introduced.

Chapter 9 – Computational Fluid Dynamics

This chapter introduces the application of commercially available computational fluid dynamics (CFD) tools available to the engineer. It introduces the fact that the strengths of these tools must be weighed against their limitations as applied by engineers to make engineering decisions.

Chapter 1 – Basic Concepts and Definitions

The basic foundations from classical dynamics and thermodynamics are reviewed and applied to air and aerodynamics. Dimensional analysis is described and the results in aerodynamics derived and discussed. The loads acting on an aircraft—forces and moments (torques)—are defined and the physical processes leading to these loads are introduced. Airfoil and wing geometry and definitions are introduced for use throughout the rest of the book.

Chapter 8 – Airfoils and Wings in Compressible Flow

In general, the equations of motion are non-linear in form and not amenable to analytical solution. Special approximate approaches exist for pure subsonic or supersonic flows. For example, the assumption of small perturbations to the freestream flow can be exploited to obtain approximate analytical solutions for both subsonic and supersonic flows around wings. Other approximate methods are also explored. Linearized compressible flow theory is used to explore the practical benefits of wing speed for high speed flight. The flight speed called the critical Mach number and methods of estimating it are found based on the results of subsonic small linearized flow. Compressible flow around wings of finite span is discussed. Wave drag, an important aerodynamic phenomenon unique to supersonic flight, is discussed and studied in examples of thin airfoils in compressible flows.

Chapter 7 – Wing Theory

Important results from potential flow and thin airfoil theory are applied to three-dimensional incompressible flow around a planar wing. The flow and loads on a single line vortex are studied and then extended to a simple horseshoe vortex. Horseshoe vortices are combined to develop Prandtls lifting line theory. The theory is then used to determine the loads on wings and how wing design choices affect such loads. Effects of aspect ratio, wing twist, varying airfoil section, taper, and similar are described by lifting line theory. Computational methods to extend beyond classical analysis are introduced.

Thin Airfoil Theory

This chapter examines the potential- or ideal-flow theory of low-speed airfoils. The application of potential-flow theory to solve for the pressure distribution on airfoils (or wing sections) is described in detail. The application of the vortex solution of the Laplace equation is applied to simulate certain effects of real flows. The result is a powerful but elementary airfoil theory capable of wide exploitation. The design of an airfoil to support a specified lift by applying camber and angle of attack is discussed. How circulation theory via the application of distributions of vortex singularities is applied to model the effects of camber and angle on airfoil performance is described. The theory is applied to examine several aerodynamic problems including the flapped airfoil and the application of a jet flap. A computational method to examine arbitrary shaped airfoils that is based on the application of a surface distribution of singularities is introduced.